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Within the realm of material science and mechanical engineering, small deformation refers to a scenario in which the shape change of a material is minimal in comparison to its overall size. To envision this, imagine a rubber band being stretched just slightly—its length increases, but not substantially relative to its original length. This concept is crucial when analyzing material behavior because it allows us to make simplifications to complex equations that describe deformation.

From the exercise, we're given a deformation condition in which \( \alpha \ll L \) indicates that the additional length caused by stretching \( \alpha / L \cdot X_2 \) is much smaller compared to the original length \( X_1 \). This constitutes the small deformation assumption, allowing us to apply linear elasticity theory to predict how materials will deform when subjected to stress. In practice, this simplification is particularly useful for calculating the infinitesimal strain tensor—a measure of deformation that reflects material strain without needing to account for more complex, non-linear effects seen in large deformations.

The displacement gradient is a tensor that emerges from the gradient of the deformation vector with respect to the original (undeformed) coordinates. Imagine you've marked a grid on a piece of stretchy material; the displacement gradient describes how each point in that grid moves and distorts as you stretch the material. Mathematically, it is represented by the derivative of the displacement vector with respect to the original coordinates.

From the exercise, we identify that the displacement components are \( u_1 = (\alpha / L) X_2, u_2 = 0, u_3 = 0 \). The displacement gradient allows us to calculate the exact changes at each point in the material and subsequently, the infinitesimal strain tensor components by examining the symmetric part of this gradient. This process is akin to zooming in on a tiny segment of the grid on our stretchy material and noting precisely how that segment has shifted or sheared. The displacement gradient forms the bedrock for determining strain within a body under the presumption of continuity and small deformations.

The term strain components pertains to the individual elements of the strain tensor, which quantitatively describe the deformation of a material. These components can denote normal strains, which represent changes in length along principal directions, or shear strains, which reflect changes in the angle between two material lines. Think of it like this: when pulling on opposite ends of a stress ball, the normal strain tells us how much the ball elongates, while the shear strain informs us about the angle changes within the ball's material.

In the given exercise, after calculating the displacement gradient, we determined that the only non-zero components of the strain tensor are the shear components \( E_{12} = E_{21} = \frac{\alpha}{2L} \). This signifies that the material is experiencing shear strain without any normal (elongational) strain. The series expansions provided in parts (b) and (c) of the exercise validate the calculated strain components by illustrating how the length and angle changes within a deformed body correlate to the determined values of strain, reinforcing our understanding of strain in the context of small deformations.